Twisted Math and Beautiful Geometry

Spira Mirabilis

Of the numerous mathematical curves we encounter in art, geometry, and nature, perhaps none can match the exquisite elegance of the logarithmic spiral. This famous curve appears, with remarkable precision, in the shape of a nautilus shell, in the horns of an antelope, and in the seed arrangements of a sunflower. It is also the ornamental motif of countless artistic designs, from antiquity to modern times. It was a favorite curve of the Dutch artist M. C. Escher (1898–1972), who used it in some of his most beautiful works, such as
Path of Life II
.

The logarithmic spiral is best described by its polar equation, written in the form
r = e
a
θ
, where
r
is the distance from the spiral’s center
O
(the “pole”) to any point
P
on the curve, θ is the angle between line
OP
and the
x-
axis,
a
is a constant that determines the spiral’s rate of growth, and
e
is the base of natural logarithms. Put differently, if we increase θ arithmetically (that is, in equal amounts),
r
will increase geometrically (in a constant ratio).

The many intriguing aspects of the logarithmic spiral all derive from this single feature. For example, a straight line from the pole
O
to any point on the spiral intercepts it at a constant angle α. For this reason, the curve is also known as an equiangular spiral. As a consequence, any sector with given angular width Δθ is similar to any other sector with the same angular width, regardless of how large or small it is. This property is manifested beautifully in the nautilus shell (
left
). The snail residing inside the shell gradually relocates from one chamber to the next, slightly larger chamber, yet all chambers are exactly similar to one another: A single blueprint serves them all.

The logarithmic spiral has been known since ancient times, but it was the Swiss mathematician Jakob Bernoulli (1654–1705) who discovered most of its properties. Bernoulli was the senior member of an eminent dynasty of mathematicians hailing from the town of Basel. He was so enamored with the logarithmic spiral that he dubbed it “spira mirabilis” and ordered it to be engraved on his tombstone after his death. His wish was fulfilled, though not quite as he had intended: For some reason, the mason engraved a linear spiral instead of a logarithmic one. (In a linear spiral the distance from the center increases arithmetically—that is, in equal amounts—as in the grooves of a vinyl record.) The linear spiral on Bernoulli’s headstone can still be seen at the cloisters of the Basel Münster, perched high on a steep hill overlooking the Rhine River.

But if a wrong spiral was engraved on Bernoulli’s tombstone, at least the inscription around it holds true:
Eadem mutata resurgo—
“Though changed, I shall arise the same.” The verse summarizes the many features of this unique curve. Stretch it, rotate it, or invert it, it always stays the same.

Notes

This angle is determined by the constant
a
; in fact, α = cot
-1
a
. In the special case when
a
= 0, we have α = 90° and the spiral becomes the unit circle
r
=
e
0
= 1. For negative values of
a
, the spiral changes its orientation from counterclockwise to clockwise as θ increases.

For more on the logarithmic spiral, see Maor,
e: The Story of a Number,
chapter 11.